Approximability and inapproximability of the star p-hub center problem with parameterized triangle inequality

A complete weighted graph G=(V,E,w) is called Δβ-metric, for some β≥1/2, if G satisfies the β-triangle inequality, i.e.,w(u,v)≤β⋅(w(u,x)+w(x,v)) for all vertices u,v,x∈V. Given a Δβ-metric graph G=(V,E,w) and a center c∈V, and an integer p, the Δβ-Starp-Hub Center problem (Δβ-SpHCP) is to find a depth-2 spanning tree T of G rooted at c such that c has exactly p children (also called hubs) and the diameter of T is minimized. In this paper, we study Δβ-SpHCP for all β≥12. We show that for any ϵ>0, to approximate the Δβ-SpHCP to a ratio g(β)−ϵ is NP-hard and give r(β)-approximation algorithms for the same problem where g(β) and r(β) are functions of β. A subclass of metric graphs is identified that Δβ-SpHCP is polynomial-time solvable. Moreover, some r(β)-approximation algorithms given in this paper meet approximation lower bounds.